Problem 49 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\tan t \cos t$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\tan t \cos t\). Answer: \(\sin t\)

Step 1: Write the expression using the identity for the tangent function

Using the identity \(\tan t=\frac{\sin t}{\cos t}\), we can rewrite the given expression as: $$\frac{\sin t}{\cos t} \cdot \cos t$$

Step 2: Multiply and simplify the expression

Now, multiply the expression and notice that \(\cos t\) in the numerator and denominator will cancel out. So, we get: $$\frac{\sin t}{\cos t} \cdot \cos t = \sin t$$ Therefore, the simplified expression is: $$\tan t \cos t = \sin t$$

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions used in mathematics. It is often represented as \( \tan t \), where \( t \) is the angle. It relates the sine and cosine functions through the identity:

• \( \tan t = \frac{\sin t}{\cos t} \)

This identity is crucial when simplifying trigonometric expressions because it allows us to express the tangent in terms of the more familiar sine and cosine functions.
By doing so, you can manipulate trigonometric expressions more flexibly. In the exercise, understanding \( \tan t \) as \( \frac{\sin t}{\cos t} \) was essential to simplify the expression \( \tan t \cos t \) into a simpler form.
This highlights how well-known identities can be used to transform and simplify mathematical expressions efficiently.

Sine Function

The sine function is a foundational trigonometric function in mathematics and is typically denoted as \( \sin t \), where \( t \) is an angle. It is a crucial part of the identity equation for the tangent function as mentioned earlier, \( \tan t = \frac{\sin t}{\cos t} \).

• \( \sin t \) measures the opposite side over the hypotenuse in a right triangle.
• It wraps the circular distance traveled on a unit circle when moving from the positive x-axis.

In the context of simplifying trigonometric expressions like \( \tan t \cos t \), the sine function appears both directly and indirectly.
The ultimate goal in the exercise was to make \( \sin t \) more prominent, highlighting its role and showing its simplicity compared to more complex trigonometric expressions.Recognizing when \( \sin t \) is useful in simplifying expressions saves time and reduces calculation complexity.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using known identities and algebraic manipulation to reduce expressions to their simplest form.

• It often makes calculations easier and solutions clearer.
• Utilizes identities like \( \tan t = \frac{\sin t}{\cos t} \) and others.

The key is to recognize which identities apply and how terms can be cancelled or combined. For example, in the expression \( \tan t \cos t \), rewriting \( \tan t \) in terms of \( \sin t \) and \( \cos t \) led to cancellation of \( \cos t \) in the numerator and denominator.
This logical step rapidly simplifies the expression from something potentially complex to simply \( \sin t \). By practicing such simplifications, students learn to see beyond the initial complexity of expressions to the straightforward core that often lies within. Brevity and clarity achieved through simplification can lend incredible clarity to both homework problems and exam questions alike.